Degree functions and projectively full ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Let (R,m)(R,m) be a 2-dimensional rational singularity with algebraically closed residue field and for which the associated graded ring is an integrally closed domain. According to Göhner, (R,m)(R,m) satisfies condition (N)(N): given a prime divisor vv, there exists a unique complete mm-primary ideal AvAv in RR with T(Av)={v}T(Av)={v} and such that any complete mm-primary ideal with unique Rees valuation vv, is a power of AvAv. We use the theory of degree functions developed by Rees and Sharp as well as some results about regular local rings, to investigate the degree coefficients d(Av,v)d(Av,v). As an immediate corollary, we find that for a simple complete m1m1-primary ideal I1I1 in an immediate quadratic transform (R1,m1)(R1,m1) of (R,m)(R,m); the inverse transform of I1I1 in RR is projectively full.